Question: $\sum\limits_{n=1}^{\infty}5^{-n}$ When applying the integral test, we get a limit that determines whether the series converges or diverges. What is this limit? Choose 1 answer: Choose 1 answer: (Choice A) A $\lim_{b\to\infty} \dfrac{5^{-b}}{\ln{(5)}}$ (Choice B) B $\lim_{b\to\infty} \dfrac{5^{b-1}}{b-1}$ (Choice C) C $\lim_{b\to\infty}\left[ 5^{-b} - \dfrac{1}{5}\right]$ (Choice D) D $\lim_{b\to\infty}\left[\dfrac{1}{5\ln{(5)}}-\dfrac{1}{5^b\ln{(5)}}\right]$
Solution: $5^{-n}$ satisfies the conditions for the integral test. This means that $\sum\limits_{n=1}^{\infty} 5^{-n}$ converges/diverges together with $\int_1^{\infty} 5^{-x}\,dx$. $\int_1^{\infty} 5^{-x}\,dx=\lim_{b\to\infty}\left[\dfrac{1}{5\ln(5)}-\dfrac{1}{5^b\ln(5)}\right]$ In conclusion, the limit that determines whether the series converges or diverges is $\lim_{b\to\infty}\left[\dfrac{1}{5\ln(5)}-\dfrac{1}{5^b\ln(5)}\right]$.